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Decimal Expansions of Fractions: Unit Fractions and Proper Fractions - Prof. Keith Conrad, Study notes of Number Theory

University of Connecticut (UConn) - Avery PointNumber Theory
Prof. Keith Conrad

The decimal expansions of unit fractions 1/b and proper fractions with a common denominator up to prime numbers 113. The table includes the decimal expansion and periodicity length for each fraction.

Typology: Study notes

Pre 2010

Uploaded on 09/24/2009

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DECIMAL DATA
The first two tables below give the decimal expansion of 1/b for 2 ≤b≤90. Some expansions do
not have their periodic part at the beginning (e.g., 1/44 = .0227 = .0227272727 . . . has an initial 02
which is not in the periodic part). Those 1/b having a purely periodic decimal expansion (that is,
it repeats right from the start) are printed in bold. A finite decimal expansion could be viewed as
an expansion with periodic part 0 (e.g., 1/16 = .0625 = .06250).
Fraction Decimal Expansion
1/2 .5
1/3 .3
1/4 .25
1/5 .20
1/6 .16
1/7 .142857
1/8 .125
1/9 .1
1/10 .1
1/11 .09
1/12 .083
1/13 .076923
1/14 .0714528
1/15 .06
1/16 .0625
1/17 .0588235294117647
1/18 .05
1/19 .052631578947368421
1/20 .05
1/21 .047619
1/22 .045
1/23 .0434782608695652173913
1/24 .0416
1/25 .04
1/26 .0384615
1/27 .037
1/28 .03571428
1/29 .0344827586206896551724137931
1/30 .03
1/31 .032258064516129
1/32 .03125
1/33 .03
1/34 .02941176470588235
1/35 .0285714
1/36 .027
1/37 .027
1/38 .0263157894736842105
1/39 .025641
1/40 .025
pf3
pf4

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Download Decimal Expansions of Fractions: Unit Fractions and Proper Fractions - Prof. Keith Conrad and more Study notes Number Theory in PDF only on Docsity!

The first two tables below give the decimal expansion of 1/b for 2 ≤ b ≤ 90. Some expansions do

which is not in the periodic part). Those 1/b having a purely periodic decimal expansion (that is, it repeats right from the start) are printed in bold. A finite decimal expansion could be viewed as an expansion with periodic part 0 (e.g., 1/16 = .0625 = .06250).

  • not have their periodic part at the beginning (e.g., 1/44 = .0227 = 0227272727 has an initial - 1/2. Fraction Decimal Expansion - 1/3 - 1/4 - 1/5. - 1/6 - 1/7. - 1/8 - 1/9 - 1/10.
    • 1/11
      • 1/12
    • 1/13
      • 1/14
      • 1/15
      • 1/16
    • 1/17
      • 1/18
    • 1/19
      • 1/20
    • 1/21
      • 1/22
    • 1/23
      • 1/24
      • 1/25
      • 1/26
    • 1/27
      • 1/28
    • 1/29
      • 1/30
    • 1/31
      • 1/32.
    • 1/33
      • 1/34
      • 1/35
      • 1/36
    • 1/37
      • 1/38
    • 1/39
      • 1/40
  • 1/41 Fraction Decimal Expansion
    • 1/42
  • 1/43
    • 1/44
    • 1/45
    • 1/46
  • 1/47
    • 1/48
  • 1/49
    • 1/50
  • 1/51
    • 1/52
  • 1/53
    • 1/54
    • 1/55
    • 1/56
  • 1/57
    • 1/58
  • 1/59
    • 1/60
  • 1/61
    • 1/62
  • 1/63
    • 1/64
    • 1/65
    • 1/66
  • 1/67
    • 1/68
  • 1/69
    • 1/70
  • 1/71
    • 1/72
  • 1/73
    • 1/74
    • 1/75
    • 1/76
  • 1/77
    • 1/78
  • 1/79
    • 1/80
  • 1/81
    • 1/82
  • 1/83
    • 1/84
    • 1/85
    • 1/86
  • 1/87
    • 1/88
  • 1/89
    • 1/90
    • p to 113, other than 2 and 5.
  • length(1/p)
    • p
  • length(1/p) - 1/3 3 2/3 Fraction Decimal Fraction Decimal - 1/7 142857 2/7 - 3/7 428571 4/7 - 5/7 714285 6/7 - 1/9 1 2/9 - 4/9 4 5/9 - 7/9 7 8/9 - 1/11 09 2/11 - 3/11 27 4/11 - 5/11 45 6/11 - 7/11 63 8/11 - 9/11 81 10/11 - 1/13 076923 2/13 - 3/13 230769 4/13 - 5/13 384615 6/13 - 7/13 538461 8/13 - 9/13 692307 10/13 - 11/13 846153 12/13 - 1/17 0588235294117647 2/17 - 3/17 1764705882352941 4/17 - 5/17 2941176470588235 6/17 - 7/17 4117647058823529 8/17 - 9/17 5294117647058823 10/17 - 11/17 6470588235294117 12/17 - 13/17 7647058823529411 14/17 - 15/17 8823529411764705 16/17 - 1/19 052631578947368421 2/19 - 3/19 157894736842105263 4/19 - 5/19 263157894736842105 6/19 - 7/19 368421052631578947 8/19 - 9/19 473684210526315789 10/19 - 11/19 578947368421052631 12/19 - 13/19 684210526315789473 14/19 - 15/19 789473684210526315 16/19 - 17/19 894736842105263157 18/19
    • 1/21 047619 2/21 Fraction Decimal Fraction Decimal
    • 4/21 190476 5/21
    • 8/21 380952 10/21
  • 11/21 523809 13/21
  • 16/21 761904 17/21
  • 19/21 904761 20/21
    • 1/23 .0434782608695652173913 2/23.
    • 3/23 .1304347826086956521739 4/23.
    • 5/23 .2173913043478260869565 6/23.
    • 7/23 .3043478260869565217391 8/23.
    • 9/23 .3913043478260869565217 10/23.
  • 11/23 .4782608695652173913043 12/23.
  • 13/23 .5652173913043478260869 14/23.
  • 15/23 .6521739130434782608695 16/23.
  • 17/23 .7391304347826086956521 18/23.
  • 19/23 .8260869565217391304347 20/23.
  • 21/23 .9130434782608695652173 22/23.
    • 1/27 037 2/27
    • 4/27 148 5/27
    • 7/27 259 8/27
  • 10/27 370 11/27
  • 13/27 481 14/27
  • 16/27 592 17/27
  • 19/27 703 20/27
  • 22/27 814 23/27
  • 25/27 925 26/27